Quantum dynamical Yang-Baxter equation over a 2 0 nonabelian base 0 2 n PING XU ∗ a J Department of Mathematics 2 The Pennsylvania State University 1 University Park, PA 16802, USA email: [email protected] ] A Q . h t Abstract a m In this paper we consider dynamical r-matricesover a nonabelianbase. There are two main [ results. First, corresponding to a fat reductive decomposition of a Lie algebra g = h ⊕ m, we construct geometrically a non-degenerate triangular dynamical r-matrix using symplectic 2 fibrations. Second, we prove that a triangular dynamical r-matrix r : h∗ −→ ∧2g corresponds v 1 to a Poisson manifold h∗×G. A special type of quantizations of this Poisson manifold, called 7 compatible star products in this paper, yields a generalized version of the quantum dynamical 0 Yang-Baxterequation(orGervais-Neveu-Felderequation). Asaresult,thequantizationproblem 4 of a general dynamical r-matrix is proposed. 0 1 0 / 1 Introduction h t a m Recently,therehasbeengrowinginterestinthesocalledquantumdynamicalYang-Baxterequation: v: R (λ)R (λ+~h(2))R (λ) = R (λ+~h(1))R (λ)R (λ+~h(3)). (1) 12 13 23 23 13 12 i X This equation arises naturally from various contexts in mathematical physics. It first appeared in r a the work of Gervais-Neveu in their study of quantum Liouville theory [24]. Recently it reappeared in Felder’s work on the quantum Knizhnik-Zamolodchikov-Bernard equation [23]. It also has been found to be connected with the quantum Caloger-Moser systems [4]. As the quantum Yang-Baxter equation is connected with quantum groups, the quantum dynamical Yang-Baxter equation is known to be connected with elliptic quantum groups [23], as well as with Hopf algebroids or quantum groupoids [20, 32, 33]. The classical counterpart of the quantum dynamical Yang-Baxter equation was first considered by Felder [23], and then studied by Etingof and Varchenko [19]. This is the so called classical dynamical Yang-Baxter equation, and a solution to such an equation (plus some other reasonable conditions) is called a classical dynamical r-matrix. More precisely, given a Lie algebra g over R (or over C) with an Abelian Lie subalgebra h, a classical dynamical r-matrix is a smooth (or meromorphic) function r : h∗ −→ g⊗g satisfying the following conditions: ∗Research partially supported byNSFgrant DMS00-72171. 1 (i). (zero weight condition) [h⊗1+1⊗h,r(λ)] = 0, ∀h∈ h; (ii). (normal condition) r +r = Ω, where Ω ∈ (S2g)g is a Casimir element; 12 21 (iii). (classical dynamical Yang-Baxter equation1) Alt(dr) − ([r ,r ]+[r ,r ]+[r ,r ]) = 0, (2) 12 13 12 23 13 23 where Altdr = (h(1)∂r23 −h(2)∂r13 +h(3)∂r12). P i ∂λi i ∂λi i ∂λi A fundamental question is whether a classical dynamical r-matrix is always quantizable. There has appeared a lot of work in this direction, for example, see [2, 25, 18]. In the triangular case (i.e., r is skew-symmetric: r (λ)+r (λ) = 0), a general quantization scheme was developed by 12 21 the author using the Fedosov method, which works for a vast class of dynamical r-matrices, called splittable triangular dynamical r-matrices [34]. Recently, Etingof and Nikshych, using the vertex- IRF transformation method, proved the existence of quantizations for the so called completely degenerate triangular dynamical r-matrices [21]. Interestingly, although the quantum dynamical Yang-Baxter equation in [23] only makes sense whenthebaseLiealgebrahisAbelian, itsclassical counterpartadmitsanimmediategeneralization for any base Lie algebra h which is not necessarily Abelian. Indeed, all one needs to do is to change the first condition (i) to: (i’). r : h∗ −→ g⊗g is H-equivariant, where H acts on h∗ by coadjoint action and on g⊗g by adjoint action. There exist many examples of such classical dynamical r-matrices. For instance, when g is a simple Lie algebra and h is a reductive Lie subalgebra containing the Cartan subalgebra, there is a classification due to Etingof-Varchenko [19]. In particular, when h = g, an explicit formula was discovered by Alekseev andMeinrenken in theirstudyof non-commutative Weil algebras [1]. Later, this was generalized by Etingof and Schiffermann [17] to a more general context. Moreover, under some regularity condition, they showed that the moduli space of dynamical r-matrices essentially consists of a single point once the initial value of the dynamical r-matrices is fixed. A natural question arises as to what should be the quantum counterpart of these r-matrices. And more generally, is any classical dynamical r-matrix (with nonabelian base) quantizable? A basic question is what the quantum dynamical Yang-Baxter equation should look like when h is nonabelian. In this paper, as a toy model, we consider the special case of triangular dynamical r-matrices andtheir quantizations. AsintheAbeliancase, theser-matrices naturally correspondto some invariant Poisson structures on h∗×G. It is standardthat quantizations of Poisson structures correspondto star products[8]. Thespecial form of the Poisson bracket relation on h∗×Gsuggests a specific form that their star products should take. This leads to our definition of compatible star products. The compatibility condition (which, in this case, is just the associativity) naturally leads to a quantum dynamical Yang-Baxter equation: Equation (33). As we shall see, this equation indeed resembles the usual quantum dynamical Yang-Baxter equation (unsymmetrized version). 1Throughout thepaper,wefollow thesign conventionin[4]for thedefinitionofaclassical dynamicalr-matrixin order to be consistent with the quantum dynamical Yang-Baxter equation (1). This differs in a sign from the one used in [19]. 2 The only difference is that the usual pointwise multiplication on C∞(h∗) is replaced by the PBW- star product, which is indeed the deformation quantization of the canonical Lie-Poisson structure on h∗. Although Equation (33) is derived by considering triangular dynamical r-matrices, it makes perfect sense for non-triangular ones as well. This naturally leads to our definition of quantization of dynamical r-matrices over an arbitrary base Lie subalgebra which is not necessary Abelian. The problem is that such an equation only makes sense for R :h∗ −→ Ug⊗Ug[[~]]. In the Abelian case, it appears that one may consider R valued in a deformed universal enveloping algebra U~g, but in most cases U~g is isomorphic to Ug[[~]] as an algebra. So Equation (33), in a certain sense, is general enough to include all the interesting cases. However, the physical meaning of this equation remains mysterious. Another main result of the paper is to give a geometric construction of triangular dynamical r-matrices. More precisely, we give an explicit construction of a triangular dynamical r-matrix from a fat reductive decomposition of a Lie algebra g = h⊕m (see Section 2 for the definition). This includes those examples of triangular dynamical r-matrices considered in [19]. Our main purpose is to show that triangular dynamical r-matrices (with nonabelian base) do rise naturally from symplectic geometry. This gives us another reason why it is important to consider their quantizations. Discussion on this part occupies Section 2. Section 3 is devoted to the discussion of compatible star products, whose associativity leads to a “twisted-cocycle” condition. In Section 4, we will derive the quantum dynamical Yang-Baxter equation from this twisted-cocycle condition. The last section contains some concluding remarks and open questions. Finally, we note that in this paper, by a dynamical r-matrix, we always mean a dynamical r- matrix over a general base Lie subalgebra unless specified. Also Lie algebras are normally assumed to be over R, although most results can be easily modified for complex Lie algebras. Acknowledgments. The author would like to thank Philip Boalch, Pavel Etingof, Boris Tsygan and David Vogan for fruitful discussions and comments. He is especially grateful to Pavel Etingof for explaining the paper [17], which inspired his interest on this topic. He also wishes to thankSimoneGuttandStefanWaldmannforprovidinghimsomeusefulreferencesonstarproducts of cotangent symplectic manifolds. 2 Classical dynamical r-matrices Inthissection, wewillgiveageometricconstructionoftriangulardynamicalr-matrices. Asweshall see, these r-matrices do arise naturally from symplectic geometry. We will show some interesting examples, which include triangular dynamical r-matrices for simple Lie algebras constructed by Etingof-Varchenko [19]. Below let us recall the definition of a classical triangular dynamical r-matrix. Let g be a Lie algebra over R (or C) and h ⊂ g a Lie subalgebra. A classical dynamical r-matrix r : h∗ −→ g⊗g is said to be triangular if it is skew symmetric: r +r = 0. In other words, a classical triangular 12 21 dynamical r-matrix is a smooth function (or meromorphic function in the complex case) r :h∗ −→ ∧2g such that (i). r : h∗ −→ ∧2g is H-equivariant, where H acts on h∗ by coadjoint action and acts on ∧2g by adjoint action. 3 (ii). ∂r 1 h ∧ − [r,r] = 0, (3) X i ∂λi 2 i where the bracket [·,·] refers to the Schouten type bracket: ∧kg⊗∧l g −→ ∧k+l−1g induced from the Lie algebra bracket on g, {h ,···,h } is a basis of h, and (λ1,···,λl) its induced coordinate 1 l system on h∗. The following proposition gives an alternative description of a classical triangular dynamical r-matrix. Proposition 2.1 A smooth function r :h∗ −→ ∧2g is a triangular dynamical r-matrix iff ∂ −→ −−→ π = πh∗ + ∧ hi +r(λ) X∂λ i i is a Poisson tensor on M = h∗ ×G, where πh∗ denotes the standard Lie (also known as Kirillov- −→ Kostant) Poisson tensor on the Lie algebra dual h∗, h ∈ X(M) is the left invariant vector field i −−→ on M generated by h ∈ h, and similarly r(λ) ∈ Γ(∧2TM) is the left invariant bivector field on M i corresponding to r(λ). Proof. Set ∂ −→ π1 = πh∗ +X∂λ ∧ hi. i i −−→ Then π = π + r(λ). Note that, for any (λ,x), π | is tangent to h∗ × xH, on which it is 1 1 (λ,x) isomorphic to the standard Poisson (symplectic) structure on the cotangent bundle T∗H (see, e.g., [27]). Here T∗H is identified with h∗×H (hence with h∗×xH) via left translations. It thus follows that [π ,π ] = 0. Therefore 1 1 −−→ −−→ −−→ [π,π] = 2[π ,r(λ)]+[r(λ),r(λ)]. 1 Now −−→ [π , r(λ)] 1 −−→ ∂ −→ −−→ = [πh∗, r(λ)]+X[∂λ ∧ hi, r(λ)] i i −−→ −−→ ∂ −→ ∂ −−→ −→ = [πh∗, r(λ)]+X[r(λ), ∂λ ]∧ hi −X ∂λ ∧[r(λ), hi]. i i i i Hence [π,π] =I +I , where 1 2 −−→ ∂ −→ −−→ −−→ I = 2 [r(λ), ]∧ h +[r(λ),r(λ)], and 1 X ∂λ i i i −−→ ∂ −−→ −→ I2 = 2[πh∗,r(λ)]−2X ∂λ ∧[r(λ),hi]. i i 4 With respect to the natural bigrading on ∧3T(h∗ × G), I and I correspond to the (0,3) and 1 2 (1,2)-terms of [π,π], respectively. It thus follows that [π,π] =0 iff I = 0 and I = 0. 1 2 It is simple to see that −→ −→ ∂ r −−−−−−−→ I =−2 h ∧ +[r(λ),r(λ)]. 1 X i ∂λi i Hence I = 0 is equivalent to Equation (3). 1 To find out the meaning of I2 = 0, let us write πh∗ = 21Pijfij(λ)∂∂λi ∧ ∂∂λj (fij = −fji). A simple computation yields that −→ ∂ ∂ r ∂ −−−−−→ I = 2 ∧ f (λ) +2 ∧[h ,r(λ)]. 2 X ∂λ X ij ∂λj X ∂λ i i i i j i Thus I = 0 is equivalent to 2 ∂r(λ) d [hi,r(λ)] = −Xfij(λ) ∂λj = dt(cid:12)(cid:12) r(Ad∗exp−1thiλ), ∀i, j (cid:12)t=0 (cid:12) which exactly means that r is H-equivariant. This concludes the proof. 2 Remark. Note that M (= h∗×G) admits a left G-action and a right H-action defined as follows: ∀(λ,x) ∈ h∗×G, y·(λ,x) = (λ,yx), ∀y ∈ G; (λ,x)·y = (Ad∗λ,xy), ∀y ∈ H. y It is clear that the Poisson structure π is invariant under both actions. And, in short, we will say that π is G×H-invariant. Definition 2.2 A classical triangular dynamical r-matrix r : h∗ −→ ∧2g is said to be non- degenerate if the corresponding Poisson structure π on M is non-degenerate, i.e., symplectic. In what follows, we will give a geometric construction of non-degenerate dynamical r-matrices. To this end, let us first recall a useful construction of a symplectic manifold from a fat principal bundle[26,31]. AprincipalbundleP(M,H) withaconnection iscalled fatonanopensubmanifold U ⊆ h∗ if the scalar-valued forms < λ,Ω > is non-degenerate on each horizontal space in TP for λ ∈ U. Here Ω is the curvature form, which is a tensorial form of type Ad on P (i.e., it is H horizontal, h-valued, and Ad -equivariant). H Given a fat bundle P(M,H) with a connection, one has a decomposition of the tangent bundle TP = Vert(P)⊕Hor(P). We may identify Vert(P) with a trivial bundle with fiber h. Thus Vert∗P ∼= h∗×P. 5 On the other hand, Vert∗P ∼= Hor⊥(P) ⊂ T∗P. Thus, by pulling back the canonical symplectic structure on T∗P, one can equip Vert∗P, hence h∗ ×P, an H-invariant presymplectic structure, where H acts on h∗ ×P by (λ,x)·h = (Ad∗λ,x·h), ∀h ∈ H and (λ,x) ∈ h∗ ×P. If U ⊆ h∗ is h an open submanifold on which P(M,H) is fat, then we obtain an H-invariant symplectic manifold U ×P. In fact, the presymplectic form ω can be described explicitly. Note that Vert∗P admits a natural fibration with T∗H being the fibers, and the connection on P induces a connection on this fiber bundle. In other words, Vert∗P is a symplectic fibration in the sense of Guillemin-Lerman- Sternberg [26]. At any point (λ,x) ∈ h∗×P ∼= Vert∗P, the presymplectic form ω can be described as follows: it restricts to the canonical two-form on the fiber; the vertical subspace is ω-orthogonal to the horizontal subspace; and the horizontal subspace is isomorphic to the horizontal subspace of T P and the restriction of ω to this subspace is the two form − < λ,Ω(x) > obtained by pairing x the curvature form with λ (see Examples 2.2-2.3 in [26]). Now assume that g = h⊕m (4) is a reductive decomposition of a Lie algebra g, i.e., h is a Lie subalgebra and m is stable under the adjoint action of h: [h, m] ⊂ m. By G, we denote a Lie group with Lie algebra g, and H the Lie subgroup corresponding to h. It is standard [28] that the decomposition (4) induces a left G-invariant connection on the principal bundle G(G/H,H), where the curvature is given by Ω(X,Y) = −[X, Y] , h−component of [X, Y]∈ g. (5) h Here X and Y are arbitrary left invariant vector fields on G belonging to m. A reductive decomposition g = h⊕m is said to be fat if the corresponding principal bundle G(G/H,H) isfatonanopensubmanifoldU ⊆ h∗. Asaconsequence, afatdecompositiong = h⊕m gives rise to a G×H-invariant symplectic structure on M = U×G, where the symplectic structure is the restriction of the canonical symplectic form on T∗G. In other words, M is a symplectic submanifold of T∗G. Here the embedding U ×G ⊆ h∗ ×G −→ g∗ ×G (∼= T∗G) is given by the natural inclusion (λ,x) −→ (pr∗λ,x), where pr :g −→ h is the projection along the decomposition g = h ⊕ m. Since the symplectic structure ω on U × G is left invariant, in order to describe ω explicitly, it suffices to specify it at a point (λ,1). Now T (U × G) ∼= h∗ ⊕ g = h∗ ⊕ h ⊕ m. (λ,1) Under this identification, we have ω = ω ⊕ω , where ω ∈ Ω2(h∗ ⊕h) is the canonical symplectic 1 2 1 two-form on T∗H at the point (λ,1) ∈ h∗×H (∼= T∗H), and ω ∈ Ω2(m) is given by 2 ω (X,Y)=< λ,[X, Y] >, ∀X,Y ∈ m. 2 h Let r(λ) ∈ ∧2m be the inverse of ω , which always exists for λ ∈ U since ω is assumed to be 2 2 non-degenerate on U. It thus follows that the Poisson structure on U ×G is ∂ −→ −−→ π = πh∗ +X ∂λ ∧ hi +r(λ). i i According to Proposition 2.1, r : U −→ ∧2m ⊂ ∧2g is a non-degenerate triangular dynamical r-matrix. Thus we have proved Theorem 2.3 Assume that g = h⊕m is a reductive decomposition which is fat on an open subman- ifold U ⊆ h∗. Then the dual of the linear map ϕ : ∧2m −→ h : (X,Y) −→ [X, Y] , ∀X,Y ∈ m h 6 defines a non-degenerate triangular dynamical r-matrix r : U(⊆ h∗)−→ ∧2m ⊂ ∧2g, ∀λ ∈ U. Here m∗ is identified with m using the non-degenerate bilinear form ϕ∗(λ) ∈ ∧2m∗. It is often more useful to express r(λ) explicitly in terms of a basis. To this end, let us choose a basis {e ,···,e } of m. Let a (λ) =< λ,[e ,e ] >, i,j = 1,···,m. By (c (λ)) we denote the 1 m ij i j h ij inverse of the matrix (a (λ)), ∀λ ∈ U. Then one has ij 1 r(λ)= c (λ)e ∧e . (6) 2 X ij i j ij Remark. (i). After the completion of the first draft, we learned that a similar formula is also obtained independently by Etingof [15]. Note that this dynamical r-matrix r is always singular at 0. To remove this singularity, one needs to make a shift of the dynamical parameter λ → λ−µ. (ii). It would be interesting to compare our formula with Theorem 3 in [17]. We end this section with some examples. Example 2.1 Let g be a simple Lie algebra over C and h a Cartan subalgebra. Let g = h⊕ (g ⊕g ) α −α M α∈∆+ be the root space decomposition, where ∆ is the set of positive roots with respect to h. Take + m = ⊕ (g ⊕g ). Then g = h⊕m is clearly a reductive decomposition. Let e ∈ g and α∈∆+ α −α α α e ∈ g be dual vectors with respect to the Killing form: (e ,e ) = 1. For any λ ∈ h∗, −α −α α −α set a (λ) =< λ,[e ,e ] >, ∀α,β ∈ ∆ ∪(−∆ ). It is then clear that a (λ) = 0, whenever αβ α β h + + αβ α+β 6= 0; and a (λ) α,−α = < λ,[e ,e ] > α −α h = (λ,α)(e ,e ) α −α = (λ,α). Therefore, from Theorem 2.3 and Equation (6), it follows that 1 r(λ) = − e ∧e α −α X (λ,α) α∈∆+ is a non-degenerate triangular dynamical r-matrix, so we have recovered this standard example in [19]. 7 Example 2.2 As in the above example, let g be a simple Lie algebra over C with a fixed Cartan subalgebra h, and l a reductive Lie subalgebra containing h. There is a subset ∆(l) of ∆ such + + that l = h⊕ (g ⊕g ). α −α M α∈∆(l)+ Let ∆ = ∆ −∆(l) , ∆(l) = ∆(l) ∪(−∆(l) ), and ∆ = ∆ ∪(−∆ ), and denote by m the + + + + + + + subspace of g: m = (g ⊕g ). X α −α α∈∆+ It is simple to see that g = l⊕m is indeed a fat reductive decomposition, and therefore induces a non-degenerate triangular dynamical r-matrix r : l∗ −→ ∧2g. To describe r explicitly, we note that the dual space l∗ admits a natural decomposition l∗ = h∗⊕ (g∗ ⊕g∗ ). M α −α α∈∆(l)+ Hence any element µ ∈ l∗ can be written as µ = λ⊕⊕ ξ , where λ ∈ h∗ and ξ ∈ g∗. Let α∈∆(l) α α α a (µ)=< µ,[e ,e ] >, ∀α,β ∈∆. It is easy to see that αβ α β l (λ,α), if α+β = 0;  a (µ) = < ξ ,[e ,e ]>, if α+β = γ ∈ ∆(l); (7) αβ  γ α β 0, otherwise.  By (c (µ)), we denote the inverse matrix of (a (µ)). According to Equation (6), αβ αβ 1 r(µ) = c (µ)e ∧e 2 X αβ α β α,β∈∆ is a non-degenerate triangular dynamical r-matrix over l∗. In particular, if µ = λ ∈ h∗, it follows immediately that 1 r(λ)= − e ∧e . (8) α −α X (λ,α) α∈∆+ Equation (8) was first obtained by Etingof-Varchenko in [19]. The following example was pointed out to us by D. Vogan. Example 2.3 Let g = Rm+n ⊕ Rm+n ⊕ R be a 2(m + n)+ 1 dimensional Heisenberg Lie alge- bra and h= Rn⊕Rn⊕R its standard Heisenberg Lie subalgebra. By {p ,q ,c}, i = 1,···,n+m, i i we denote the standard generators of g and {p ,q ,c}, i = 1,···,n, the generators of h. Let m+i m+i m be the subspace of g generated by {p ,q }, i = 1,···,m. It is then clear that g = h⊕m is a i i reductive decomposition. Let {p∗,q∗,c∗}, i= 1,···,n+m, be the dual basis corresponding to the i i standard generators of g. For any λ ∈ h∗, write λ = n (a p∗ +b q∗ )+xc∗. This induces i=1 i m+i i m+i P 8 a coordinate system on h∗, and therefore a function on h∗ can be identified with a function with variables (a ,b ,x). It is clear that i i ω(p ,q )(λ) =< λ,[p ,q ] >= xδ ; i j i j h ij ω(p ,p )= ω(q ,q )= 0, ∀i, j = 1,···,m. i j i j It thus follows that m 1 r(a ,b ,x) = − p ∧q : h∗ −→ ∧2g i i i i xX i=1 is a non-degenerate triangular dynamical r-matrix. 3 Compatible star products From Proposition 2.1, we know that a triangular dynamical r-matrix r : h∗ −→ ∧2g is equivalent to a special type of Poisson structures on h∗×G. It is thus very natural to expect that quantization of r can be derived from a certain special type of star-products on h∗×G. It is simple to see that the Poisson brackets on C∞(h∗×G) can be described as follows: (i). for any f,g ∈ C∞(h∗), {f,g} = {f,g} ; πh∗ −→ (ii). for any f ∈ C∞(h∗) and g ∈ C∞(G), {f,g} = (∂f )(h g); Pi ∂λi i −−→ (iii). for any f,g ∈ C∞(G), {f,g} = r(λ)(f,g). These Poisson bracket relations naturally motivate the following: Definition 3.1 A star product ∗~ on M = h∗×G is called a compatible star product if (i). for any f,g ∈ C∞(h∗), f(λ)∗~g(λ) = f(λ)∗g(λ); (9) (ii). for any f(x)∈ C∞(G) and g(λ) ∈ C∞(h∗), f(x)∗~g(λ) = f(x)g(λ); (10) (iii). for any f(λ)∈ C∞(h∗) and g(x) ∈ C∞(G), ∞ ~k ∂kf −→ −→ f(λ)∗~g(x) = X k! ∂λi1···∂λikhi1···hikg; (11) k=0 (iv). for any f(x), g(x) ∈ C∞(G), −−−→ f(x)∗~g(x) = F(λ)(f,g), (12) where F(λ) is a smooth function F : h∗ −→ Ug⊗Ug[[~]] such that F = 1+~F +O(~2). 1 9 Here∗denotesthestandardPBW-starproductonh∗ quantizingthecanonicalLie-Poissonstructure (see [12]), whose definition is recalled below. Let h~ = h[[~]] be a Lie algebra with the Lie bracket [X,Y]~ = ~[X,Y], ∀X,Y ∈ h[[~]], and σ : S(h)[[~]] ∼= Uh~ be the Poincar´e-Birkhoff-Witt map, which is a vector space isomorphism. Thus the multiplication on Uh~ induces a multiplication on S(h)[[~]] (∼= Pol(h∗)[[~]]), hence on C∞(h∗)[[~]], which is denoted by ∗. It is easy to check that ∗ satisfies 1 f ∗g = fg+ 2~{f,g}πh∗ +X~kBk(f,g)+···, ∀f,g ∈C∞(h∗), k≥0 where B ’s are bidifferential operators. In other words, ∗ is indeed a star product on h∗, which is k called the PBW-star product. The following proposition is quite obvious. Proposition 3.2 The classical limit of a compatible star product is the Poisson structure π = −→ −−→ πh∗ +Pi ∂∂λi ∧ hi +r(λ), where r(λ) = F12(λ)−F21(λ). Below we will study some important properties of compatible star products. Proposition 3.3 A compatible star product is always invariant under the left G-action. It is right H-invariant iff F :h∗ −→ Ug⊗Ug[[~]] is H-equivariant, where H acts on h∗ by the coadjoint action and on Ug⊗Ug by the adjoint action. Proof. First of all, note that Equations (9-12) completely determine a star product. It is clear, from these equations, that ∗~ is left G-invariant. As for the right H-action, it is obvious from Equation (10) that ∗~ is invariant for f(x)∗~g(λ). It is standard that ∗ is invariant under the coadjoint action, so it follows from Equation (9) that f(λ)∗~g(λ) is also H-invariant. For any h ∈h, g(x) ∈ C∞(G) and any fixed y ∈ H, −→ h(R∗g)(x) = (L h)(R∗g) y x y = (R L h)(g) y x = (LxyAdy−1h)(g) −−−−→ = (Ady−1hg)(xy) −−−−→ = [Ry∗(Ady−1hg)](x). Thus it follows that −→ −→ −→ −→ h ···h (R∗g) = R∗(h′ ···h′ g), (13) i1 ik y y i1 ik 10